How does Pauli principle apply to the system?  For one thing, it seems to apply something like repulsion? But how to put it in a more formal way? It is probably not the same as feel the interaction at interparticle distance.  Probably more than that.  

In principle you take care of it by adopting the second quantization.  

How can you take care of at certain length scale while not other? 

Is Gross-Pitaevskii equation a way? Check out \cite{Pethick} and \cite{Leggett}
